Slices that Pie Riddle
Write-up by Craig Corsi and Steven J Miller
Below we talk about the Slices of Pie riddle. If this difficulty may not have the exact same mathematical prerequisites as some of the other riddles, it exposes a couple of preconceptions that human being often have when fixing riddles. We research these preconceptions and show how they make the solution more daunting to reach. After addressing the riddle, we will certainly look at how to discover the riddle in a much more general setup using a usmam.orgsite referred to as the online Encyclopedia of creature Sequences:http://oeis.org/ .
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Here"s the original question:
Slices that Pie: If you have a circle and you are only provided three currently to divide it in 7 pieces, how can you achieve this task?
The first step in addressing a problem is to make certain we recognize what all the words mean. We have to clarify the the �circle� in this riddle describes the whole disk, and also not just the boundary. As a nice added exercise, display you can�t division the perimeter that a circle right into seven pieces with just three lines.
Let�s look in ~ what happens v fewer present and shot to sniff out a pattern. If we have zero lines climate there�s simply one region. If there is one line then there space two regions. If there are two lines then we have the right to get 4 regions (and no more, as the ideal a line have the right to do is division each existing an ar in half). We watch a pattern: 1, 2, 4; that seems natural to conjecture that the following term is 8. Unfortunately, this sample doesn�t continue, and the problem says the the next term is 7, no 8.
So what type of preconceptions make our task seem impossible? that is, what assumptions might it is in made i m sorry don"t should (and shouldn"t) it is in made? First, we might think the all pieces of the circle have to be the very same size and also shape, together if we were cutting a pie and also giving the piece to our friends. One way to accomplish this is to make seven cuts, every of which cut halfway right into the pie, emotional the center. This offers the right variety of pieces yet with too numerous cuts. While this isn�t the solution, it does aid us obtain on the appropriate track. If a problem is hard, shot doing a simpler one first. Is over there some variety of cuts that will certainly divide our circle right into seven pieces? possibly we deserve to do that with an ext than three, and also then see ways to simplify or remove some the the cuts.
Alternatively, we can make 3 cuts v the center, separating the circle into six pieces, every whose angle equates to sixty degrees. This is the right number of cuts, yet we don"t have enough pieces. Us would require one an ext cut. Thus, we�ve shown the complying with fact:
Fact : At the very least one of the cut does not pass with the facility of the circle.
This is a significant step forward. There room so many different methods to cut the circle, we require to discover a method to test all the possibilities. This offers us a wonderful start. We�ll do one more observation before giving a solution:
Fact : every line starts at a point on the circle�s perimeter and also ends in ~ a allude on the circle�perimeter.
Proof: If not, we have the right to only create additional regions by extending a line the is either totally within the circle, or going from the perimeter come some interior point.
The objective of this monitoring is to assist us navigate all the feasible configurations. We recognize at the very least one line doesn�t pass with the facility of the circle, and the difficulty is tantamount to choosing 6 points on the one (in 3 pairs that 2) so the the result lines division the circle right into 7 pieces.
It�s an excellent to try lots of various sets of 3 cuts; this is a an excellent way to construct intuition and also get a feel for what goes wrong (and maybe allude out what we�ll have to do come make points go right). Various other attempts could include having actually two upright lines and also one horizontal line, or one vertical and two horizontal. These both provide only 6 pieces. Three parallel currently is worse, together only four pieces space created. Carry out we have to make also weirder pieces to attain our task? every one of our examples so far have had all of the pieces encompass some of the border the the circle. Is it feasible to find an example where this does not happen?
Let"s put these monitorings aside and instead shot to search for a solution much more methodically. A an excellent way to think about the difficulty is to include lines top top the circle one through one and see how countless pieces we can get from the circle after each step. Let"s it is in greedy: place each line in the best feasible place (giving us the most number of pieces) before adding the next line. This might not it is in the means to go, but it�s precious a try.
The first line can divide the circle right into at most two pieces. Adding another line can make in ~ most four pieces total. (We can"t relocate pieces the the circle between cuts!) there are many ways to perform two cuts separating the one into 4 pieces. Let�s start with the easiest possible: take 2 lines going through the facility at appropriate angles to every other. There are several other configuration we could try, however this has actually the advantage of gift simple. Let�s see what this offers us.
So, where must we location the third line? because that each the the four pieces the the circle determined by the an initial two lines, one of two people the third line divides the piece into two, or the does not. Therefore how countless of the 4 pieces can we division at once? the doesn"t look like 4 is possible, but three is really doable. In the number below, we start with 4 pieces. A thin third line divides the blue, green, and red pieces right into two, providing a full of 7 pieces, as desired.
One that my favourite features about mathematics is that generally there is an ext than one method to fix a problem, with various solutions highlighting different aspects. This difficulty is a great example, as there are at the very least two remedies which appear fundamentally different. Let�s talk around how to find an additional one.
Here"s another means to attack the problem. Pick 3 points ~ above the circle and connect the point out with three lines. This divides the circle into three caps and also a triangle, for 4 regions total. What we desire to do currently is perturb our initial three line segments, and also see exactly how that adds more regions together we relocate them.
Specifically, we slide each line just a tiny bit towards the middle of the circle, and we check out that the present overlap enough to division the circle into seven pieces. For this reason this offers us ours answer together well!
GENERALIZATIONS and the OEIS:
A huge part of mathematics is taking a systems to a specific problem and trying to generalize. Space there related concerns to ask? exactly how does a specific problem fit right into a much more general framework? What are the key features? What properties need to the remedies of the general situation have?
There are numerous ways we could generalize this problem. Probably the most natural (but by no method the only) opportunity is to ask:
How many pieces have the right to we acquire with 4 lines? With 5 lines? with n lines?
As the variety of lines increases and the number of possible configuration of present skyrockets, we begin to concern whether the approaches we supplied to resolve the original riddle will work-related in the general case. Also, nobody is daunting us to with a particular number of pieces anymore, which means that we have to guess what is and also is not possible and climate prove our claim. Proving that a certain number of pieces is impossible deserve to be complicated (maybe not for a small variety of lines top top the circle, but certainly together that number it s okay larger).
At this point we might be thinking that who else through a lot much more mathematical background has already solved this trouble using progressed mathematics and also it may instead be worth our if to try another problem. In this case, the digital Encyclopedia of integer Sequences (OEIS) might be may be to aid us research what is known about the riddle:http://oeis.org/.
This encyclopedia is amazing. If we understand the first couple of terms of part sequence that integers, this usmam.orgsite can often actually determine our sequence and continue it because that us, based on what others have already discovered. Shot inputting (i) 1, 1, 2, 3, 5; ; (iii) 1, 3, 6, 10, 15, 21; (iii) 1, 1, 2, 5, 14, 42, 132; (iv) 27, 82, 41, 124, 62, 31, 94, 47, 142; (v) 4, 11, 31, 83, 227, 616, 1674. While some of these collection are hopefully old friends, I�m hoping at least the last two are new.
Returning come our problem, we uncovered earlier that through 0, 1, 2, and also 3 cut we can consist of to 1, 2, 4, and 7 pieces.
If us input 1, 2, 4, 7, and scroll down a bit, us find:
Central polygonal number (the Lazy Caterer"s sequence): n(n+1)/2 + 1; or, maximal number of pieces created when slicing a pancake through n cuts.
So no only has actually the general trouble been solved, yet there is a yes, really nice formula for the number of slices possible with n cuts. The first couple of terms space 1, 2, 4, 7, 11, 16, 22, 29, 37, 46.
If we take a look at the sequence, we check out that the jump in between terms boosts by one each time. This renders sense. Going earlier to our discussion of how brand-new cuts take it existing pieces and cut them into two, we experienced that the an initial slice impacted one piece, the 2nd slice impacted two present pieces, and the third slice affected three pieces, and also so, as we would expect, the variety of pieces that we can reduced at a time boosts by one because that every new slice.
Another means of phrasing this is the if us look in ~ the distinction between nearby terms, we gain a brand-new sequence: 1, 2, 3, 4, 5, 6, 7, 8, 9�. This succession is much easier to understand, and also once we know it we deserve to go ago to our original sequence. This is a great lesson: once you�re offered something come study, gain data points if possible, and also look because that relations. It may be the case that a related difficulty is clearer 보다 the original.
What rather is cool about OEIS? If we look listed below a succession we gain a comments section that describes the succession in many other ways. We additionally get a number of formulas and references for additional reading which may assist place the object in a larger mathematical background.
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Finally, we finish with a few additional generalizations. If you�re to teach this in a class, avoid here and also don�t check out further. Try and come up through your very own questions. View what you students or classmates create, and then come back.readjust the geometry: instead of a circle, what if us take various shapes and use 3 lines. How critical is it the we begin with a circle? instead of making use of lines use parabolas. How countless regions deserve to we acquire with three parabolas cutting a circle? What about other curves? rise the dimension: look at a sphere, and also instead of three lines take 3 planes slicing it. What happens with more lines? What if our an ar has holes? carry out we acquire the exact same answers as we increase the variety of lines?